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C₂×ΓL(1,𝔽₈) #184

order	42
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = AGL(1,𝔽₇) (order 42) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂
holomorph	G ⋊ Aut(G) = C₂×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₂×C₇)⋊C₃

Splittings

Fitting, socle, Hall	C₂×C₇ ⋊ C₃
derived, Sylow	C₇ ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 14

A minimal faithful permutation representation — 2 generators on 9 points:

(1,2)(4,5,6)(7,9,8);(3,4,5,7,6,8,9)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	3	3	7	7	1	3	3	7	7
order	1	7	7	3	3	2	14	14	6	6
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	3	$\frac{- 1 - i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	0	0	3	$\frac{- 1 - i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	0	0
χ₅	3	$\frac{- 1 + i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	0	0	3	$\frac{- 1 + i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	0	0
χ₆	1	1	1	1	1	-1	-1	-1	-1	-1
χ₇	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	-1	-1	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₈	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	-1	-1	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₉	3	$\frac{- 1 - i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	0	0	-3	$\frac{1 + i \sqrt{7}}{2}$	$\frac{1 - i \sqrt{7}}{2}$	0	0
χ₁₀	3	$\frac{- 1 + i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	0	0	-3	$\frac{1 - i \sqrt{7}}{2}$	$\frac{1 + i \sqrt{7}}{2}$	0	0

Here i is the imaginary unit (i² = −1).

Relationships

maximal subgroup of	ΓL(1,𝔽₈)⋊C₄, C₄×ΓL(1,𝔽₈), C₂×AGL(1,𝔽₇), …
maximal normal subgroup of	ΓL(1,𝔽₈)⋊C₄, C₄×ΓL(1,𝔽₈), C₂×AGL(1,𝔽₇), …
maximal quotient of	C₄×ΓL(1,𝔽₈), C₂²×ΓL(1,𝔽₈), C₂×C₇⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(42, 2);